Combinations of six cards, given 4 of a kind, 1 pair

[F19]

Any help is appreciated!

1) A card game hand has six cards. How many hands can be formed if the hand contains a four of a kind and a pair?

For assistance:
52 cards in a deck
4 suits (hearts, diamonds, spades, clubs)
13 cards per suit (A,2,3,4,5,6,7,8,9,10,J,Q,K)

Thanks alot!

 

[mmm4444bot]

If 4-of-a-kind is dealt, then it's like having a deck with 12 cards per suit from which to choose a pair.

There are four cards from which to deal a pair of 2s. Do you know how many pairs can be formed from the four 2s?

There are four cards from which to deal a pair of 3s. Do you know how many pairs can be formed?

There are four cards from which to deal a pair of any rank, other than the rank already removed for the 4-of-a-kind.

If you need more help, then please show whatever work or reasoning that you can do so far.

 

[soroban]

Re: Combinations

Hello, F19!

Another approach . . .

1) A card game hand has six cards.
How many hands can be formed if the hand contains a four-of-a-kind and a pair?

For the four-of-a-kind, there are 13 choices of values.
. . And there is 1 way to get that four-of-a-kind.

For the pair, there are 12 choices of values.
. . And there are: [sub:1atcr0v6]4[/sub:1atcr0v6]C[sub:1atcr0v6]2[/sub:1atcr0v6] ways to get that pair.

Therefore . . .

 

[Denis]

The way the question is asked means only 312 hands.

1111:22-22 ...... 1111:KK-KK : 24
2222:11-11 ...... 2222:KK-KK : 24
...
KKKK:11-11 ...... KKKK:QQ-QQ: 24

24*13 = 312